Using the factor theorem it is found that a + b, b + c and c + a … - Vidyadip

MCQ

Using the factor theorem it is found that $a + b, b + c$ and $c + a$ are three factors of the determinant $\begin{vmatrix}-2\text{a}&\text{a}+\text{b}&\text{a}+\text{c}\\\text{b}+\text{a}&-2\text{b}&\text{b}+\text{c}\\\text{c}+\text{a}&\text{c}+\text{b}&-2\text{c}\end{vmatrix}$ the other factor in the value of the determinant is:

✓
$4$
B
$2$
C
$a + b + c$
D
None of these.

✓

Answer

Correct option: A.

$4$

$\triangle=\begin{vmatrix}-2\text{a}&\text{a}+\text{b}&\text{a}+\text{c}\\\text{b}+\text{a}&-2\text{b}&\text{b}+\text{c}\\\text{c}+\text{a}&\text{c}+\text{b}&-2\text{c}\end{vmatrix}$
Let $a + b = 2C, b + c = 2A$ and $c + a = 2B$
$\Rightarrow a + b + b + c + c + a = 2A + 2B + 2C$
$\Rightarrow 2(a + b + c) = (A + B + C)$
Also$, a = (a + b + c) - (b + c) = (A + B + C) - 2A = B + C - A$
Similarly$, b = C + A - B, c = A + B - C$
Hence$, 4$ is the order factor of the determinant.

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